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% Copyright (C) 1996, 1997, 2014 Cambridge University Press

\documentclass{jfm}
\usepackage{graphicx}
\usepackage{epstopdf, epsfig}
\usepackage{psfrag}
\usepackage{subfigure}
\usepackage{bm}% bold math
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}

\shorttitle{Why the Gaster transformation cannot be used to the fundamental resonance?}
\shortauthor{J. Xu, Z. Zhang, J. Liu}

\title{Why the traditional Gaster transformation cannot be used to the fundamental secondary instability problem?}

\author{Jiakuan Xu\aff{1},Zhongyu Zhang\aff{2},
  Jianxin Liu\aff{2}
  \corresp{\email{shookware@tju.edu.cn}}}

\affiliation{\aff{1}School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, P.R.China
\aff{2}Laboratory for High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, P.R.China}

\begin{document}

\maketitle

\begin{abstract}
	In boundary layer flows, stability analysis can usually be carried out in two ways: temporal mode and spatial mode. Spatial mode is very helpful for studying the spatial evolution characteristics of unstable disturbances. Therefore, it is very important but computationally expensive and inferior to time. Gaster proposed the Gaster transform, which is a method to obtain spatial mode solutions from time mode conversion. This method is still employed by Herbert in the problem of secondary instability, and has achieved good results in the incompressible boundary layers. However, we found that the Gaster transform is only applicable to the calculation of the growth-rate of subharmonic resonance in the secondary instability problem, and is not applicable to the fundamental resonance. This paper derives and analyzes mathematically why the Gaster transform is suitable for subharmonic resonance but not for fundamental resonance, and it is verified and discussed in different hypersonic flows.
	
	% Making the mechanism of secondary instability clear is very important for researchers to understand the turbulence generation and flow control. In this paper, we focus on the main route of secondary instability to turbulence, including the fundamental breakdown and subharmonic breakdown, especially the former one. According to the existing research papers, sometimes the subharmonic breakdown dominates the secondary instability, but sometimes fundamental breakdown does. In this article, We are trying to explain the reason for the difference and when which mode would dominate. After detailed analysis, we found that the saturated amplitude of primary Mack mode is the key to determine the type of breakdown. With further study, the authors found that the saturated amplitude of primary mode depends on the evolution history which may be affected by the initial amplitude and pressure gradient.
	% {\it Journal of Fluid Mechanics}
\end{abstract}

\begin{keywords}
	Hypersonic boundary layer, Mack mode, secondary instability, fundamental breakdown, subharmonic breakdown
\end{keywords}

\section{Introduction}
The boundary layer laminar-turbulent transition mechanism is a very significant issue in fluid mechanics. One of the crucial problems in laminar-turbulent transition is what mechanism in the transition process leads to the three dimensional(3D) turbulence structures in two dimensional(2D) laminar boundary layers. In 2D incompressible boundary layers, the least stable instability, Tollmien-Schlichting (T-S) waves instability, is still two-dimensional. How do they develop to the three-dimensional unsteady turbulence? In order to explain this problem, the secondary instability theory, proposed by \citet{herbert1988secondary}, which indicates that there should be a kind of secondary instability mechanism in the nonlinear interaction process from 2D laminar flows to 3D turbulence, is employed here. This theory is one of the methods for explaining the generation of 3D structures in turbulence. In Herbert's secondary instability theory, when the primary 2D T-S disturbances reach a threshold value, the basic flow of the instability can be seen as the sum of the 2D boundary layer and the corresponding primary mode disturbance, and then a new linear stability theory of the new basic flow, i.e., the secondary instability mechanism, can excite the growth of the infinitesimal disturbances which are three-dimensional. This linear instability problem can be solved by the Floquet theory \citep{herbert1983secondary}. With different Floquet parameters, different harmonic modes can be obtained. Consequently, these harmonic modes in secondary instability can be classified into three types that include fundamental mode, subharmonic mode and detuned mode to describe the periodic modes.

In fact, the earlier 3D structures in turbulence are predicted by secondary instability analysis in lots of cases because it can be seen as the precursor of transition. As a result, the breakdown caused by secondary instability can also be called the fundamental breakdown, subharmonic breakdown and detuned breakdown, which has been observed in many cases \citep{schmid2012stability,wu2019nonlinear}. Fundamental modes, often lead to the fundamental breakdown, having the same streamwise wavelength and frequency as the primary 2D disturbance. The fundamental breakdown process was firstly observed by Klebanoff, so it is also called K-type breakdown. The subharmonic modes have twice the streamwise wavelength and one half the frequency as the primary mode disturbance. They cause the subharmonic breakdown, i.e., H/N-type breakdown. In the subharmonic breakdown, the turbulent spots are always staggered. Detuned modes are the modes between fundamental and subharmonic ones, which can be observed in the  literature\citep{li2012secondary}.

About the investigation of the subharmonic breakdown (H/N type), triad resonance theory is firstly proposed by \citet{craik1971non} in incompressible flows, which was enhanced subsequently by \citet{volodin1978three} and \citet{volodin1981nature}. Then, Herbert's secondary instability theory based on Floquet theory mentioned above gives a reasonable explanation for subharmonic resonance \citep{herbert1988secondary}. In 1977, \citet{kachanov1977nonlinear} firstly observed the N-type subharmonic breakdown in the experiment, which was confirmed by \citet{saric1984experiments}, \citet{kachanov1984resonant} and \citet{bake2002turbulence} and summarized by \citet{kachanov1994physical}. All these studies point out that subharmonic mode usually dominates the secondary instability. Nevertheless, in high-speed flows, there are more complex routes of breakdown to turbulence, and subharmonic mode also plays a very important role in these flows. Kosinov and his
co-workers \citep{kosinov1985development,kosinov1990experiments,kosinov1994investigation,kosinov1994experiments,kosinov1997experimental} conducted lots of measurements on the subharmonic resonance mechanism in supersonic boundary layers and further investigation was performed by \citet{ermolaev1996experimental}. The corresponding theoretical analysis and numerical simulations are later carried out by \citet{kosinov1996resonance}, \citet{mayer2008investigation} and \citet{fezer2000spatial} which verified the existence and feasibility of subharmonic resonance.
However, in low-Mach number boundary layers, the typical oblique breakdown, proposed by \citet{fasel1993direct}, plays a crucial role in the secondary instability of first mode primary. A lots of direct numerical simulations (DNS) work\citep{mayer2007investigation,laible2009numerical,mayer2010detailed,laible2011numerical} were employed, which confirmed the experiment\citep{schneider1994experiments} and theoretical analysis \citep{chang1994oblique,tumin1995three,tumin1996nonlinear}. Because oblique breakdown is not the focus in this paper, it will not go into details here. With the increasing of Mach number, the second mode, i.e., the so-called Mack mode, always becomes the dominant instability in hypersonic boundary layers while the importance of the first mode becomes reduced. Different from the first mode instability, the Mack mode is always planar in the 2D flow. This characteristic is similar to that of T-S disturbance in incompressible cases, but the Mack mode instability results from the compressible effect at higher Mach number. Hence, it seems that some very different conclusions may be found in hypersonic flows. Because the planar most unstable Mack mode instability is primary, the flow is still 2D when the disturbance grows to a finite amplitude. It implies that a mechanism is required to explain the generation of the 3D structures in the turbulence. It seems that the secondary instability theory is a proper one to explain this question, just as the general understanding of the secondary instability in incompressible flows. About the subharmonic breakdown in supersonic/hypersonic flows, several studies through the secondary instability theory were performed. \citet{masad1990subharmonic} conducted the research of the effect of Mach number, spanwise wave number, primary disturbance amplitude, Reynolds number and frequencies on subharmonic instability of compressible boundary layers by the Floquet theory. Furthermore, the suction effects and wall shaping were evaluated and discussed by \citet{masad1992effects} whose study suggests that the higher Mach number stabilizes the subharmonic mode when the primary mode is the first mode while destabilizes that when the primary one is the Mack mode. \citet{el1992secondary} also studied the effects of Mach number, wall suction and cooling for secondary instability in high-speed flows. He found that the secondary subharmonic instability of an insulated wall boundary layer is weakened as Mach number increasing and the cooling wall effect destabilizes the secondary subharmonic mode of a Mack mode primary disturbance, but stabilizes it when the primary disturbance is the first mode.

In the meanwhile, the other work focused on the characteristics of secondary instability at higher Mach number. Though it is complicated to investigate the details in the wind-tunnel experiments because the quiet equipment is required, the theoretical analysis was performed at first in the earlier stage to investigate this question. In 1992, \citet{ng1992secondary} investigated the secondary instability in compressible boundary layers by the Floquet theory and pointed out that at a high Mach number of 4.5, the secondary instability of the primary Mack mode is stronger than that of the primary first mode and the subharmonic resonance dominates the instability. It seems that the conclusion in the hypersonic flows is very similar to that in the low-speed flows; however, the results in recent ten years suggest that the problem is very complex. From 2009\citep{wheaton2009instability}, Schneider's group has conducted a series of experiments on a flared cone in the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. The measurements indicate that the streaky structures can be found in the earlier stage of the transition. This characteristic suggests that the fundamental breakdown, not the subharmonic one, may dominate the secondary instability on the flared cone. A long term experimental study has also been performed and reveals a very similar conclusion\citep{schneider2015,chynoweth2018measurements}. Furthermore, about the topic of Mack mode dominated transition, in recent years, Lee's group also has brought impressive physical insights into the primary and secondary instabilities of Mack mode disturbances in hypersonic boundary layers through both experimental and theoretical methods\cite{tang2015development,chen2017interactions,zhuyd_aiaaj2016,zhuyid_pof2018,zhuyd_jfm_2018,LeeCB_2019Flow,zhuyd_pof_2020Acoustic}. The direct numerical simulation can also supply the results with quiet background noise, and their results confirm the conclusion in the experiments in the quiet hypersonic flows. Besides, more results also were obtained in numerical investigations. In 2012, \citet{sivasubramanian2012nonlinear} selected DNS method to simulate the fundamental breakdown on Purdue flared cone and found the typical ``hot" streak pattern which is similar with the experimental phenomenon. In addition to the stationary mode produced by self-interactions of the Mack mode, very low-frequency waves were observed in the process of breakdown at Mach 6.0 boundary layers \citep{li2010direct,sivasubramanian2014numerical} and the nonlinear mechanism of the interactions between the low-frequency waves and the Mack mode has been explored by \citet{chen2017interactions}. \citet{sivasubramanian2015direct} simulated the so-called ``controlled" breakdown of the fundamental resonance scenario based on the conditions of the experiments and detailed data analysis for the fundamental breakdown on the Purdue flared cone confirmed the similar ``secondary" streaks and azimuthal wave number with the measured data. In 2016 and 2017, \citet{hader2016laminar} and \citet{hader2017fundamental} demonstrated that the fundamental breakdown is initiated by the nonlinear resonance of a high amplitude 2D primary wave. A symmetric pair of low amplitude 3D secondary waves at the same frequency and the stationary mode (0,1) are the strongest secondary instability disturbances which are consistent with the measured data. \citet{meersman2018hypersonic} compared the flared with straight cone using DNS and pointed out that flared cone is more amplified than the straight cone, with another new phenomenon named ``double peak" observed for the straight cone.

In the theoretical analysis, the secondary instability was also performed more deeply for understanding it in the hypersonic boundary layers. \citet{li2012secondary} conducted the research about fundamental breakdown in hypersonic boundary layers using nonlinear parabolized stability equations(NPSE) and secondary instability analysis, which indicates that a strong fundamental secondary instability can exist for a range of initial amplitudes of the most amplified second mode disturbance in the Purdue Mach 6 flared cone while in the straight cone, the subharmonic mode dominates the secondary instability within a certain range of azimuthal wavenumbers. \citet{liu2019fundamental} also investigated the fundamental resonance in high-speed boundary layers over flat plates by the Floquet expansion and found that the fundamental family is the least stable secondary instability branch when the amplitude of the primary mode instability reaches a threshold value.

Although there are lots of work concentrated on the secondary instability problem in the hypersonic boundary layers, more work is needed in the theoretical analysis. It is well-known that the so-called K type breakdown, i.e., fundamental breakdown, was investigated in moderate and high turbulence density freestream conditions in the incompressible experiment \citep{klebanoff1962three}. However, in hypersonic boundary layers, not only the wind-tunnel experiments but also the numerical simulations both illustrate that the fundamental breakdown is observed in the lower level freestream noise. The past work has reported that the secondary instability on the flared cone is the fundamental mode but there is no secondary instability analysis to prove that this breakdown type observed in the experiments and the DNS. It is found in \citet{liu2019fundamental} that the amplitude of primary mode disturbance is the main factor in some boundary layers, but what the mechanism makes the fundamental mode dominate the secondary instability is still a problem. Why the fundamental breakdown can be observed in the hypersonic flared cone flow?
% Since it is very difficult to investigate the details in experiments, there is little measurement to validate these theoretical solutions for a long time. From 2009, Schneider's group conducted the experiments on flared cone in the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. What's more important, the measurements indicate that the fundamental breakdown dominates the secondary instability on the flared cone. Hence, the fundamental breakdown will be introduced in the following,
% In the incompressible experiment of \citet{klebanoff1962three}, the so-called 'K' type breakdown, i.e., fundamental breakdown, are investigated in moderate and high turbulence density freestream conditions. However, in the Boeing/AFOSR Mach 6 Quiet Tunnel, Purdue research group also found fundamental breakdown on the flared cone in hypersonic boundary layer at Mach 6.0 \citep{wheaton2009instability} and a long-term experimental study has been performed and summarized in \citep{schneider2015,chynoweth2018measurements}. Combined with the numerical simulations, theoretical analysis and experimental data, a more detailed and comprehensive summary published in the literature with the title of "History and Progress of Boundary-Layer Transition on a Mach-6 Flared Cone"\citep{chynoweth2019history}. In 2012, \citet{sivasubramanian2012nonlinear} selected DNS method to simulate the fundamental breakdown on Purdue flared cone and found the typical "hot" streak pattern which is similar with the experimental phenomenon. Then, Li Fei \citep{li2012secondary} conducted the research about fundamental breakdown in hypersonic boundary layers using nonlinear parabolized stability equations(NPSE) and secondary instability analysis, which indicates that a strong fundamental secondary instability can exist for a range of initial amplitudes of the most amplified second mode disturbance in the Purdue Mach 6 flared cone.   \citet{liu2019fundamental} also investigated in high-speed boundary layers over flat plates by numerical methodology and found fundamental family is the least stable secondary instability when the amplitude of the primary mode instability reaches a threshold value. In 2019, \citet{fasel2019JFM} did further Direct numerical simulations of hypersonic boundary-layer transition for a flared cone about fundamental breakdown, which provides strong evidence that the fundamental breakdown is a dominant and viable path to transition for the BAM6QT conditions. From the above, these research contents prove that if a 'controlled' dominant primary mode is introduced in the inlet of numerical simulation domain, the reasonable results in a good agreement with experimental data can be obtained. Therefore, in this paper, all the research in this paper will follow this line of thinking.
% Now, it time to make it clear what will do in this article. Through Fasel's sufficient work, it can be found that fundamental breakdown usually dominate the secondary instability with a large saturated amplitude of primary mode. In this paper, we want to demonstrate that the amplitude of primary mode is the crucial factor. Furthermore, in low-speed boundary layers, a large amplitude of freestream disturbance will result in a large saturated amplitude of primary mode. However, in hypersonic quiet wind tunnel, how to get a large primary amplitude is a question. The factor determining the saturated amplitude is the dispersion relation and evolution history of the primary disturbance wave, and we guess the pressure gradient is a very important parameter. Hence, why does the amplitude have an impact and under what circumstances a large primary amplitude occurs, what is the mechanism behind it. In this paper, we are trying to explain these questions using NPSE method, secondary instability analysis and energy balance analysis methods.

The questions above will be answered in this paper. In the following content, \S\ref{sec:mathematical} will give detailed information used for the present study.  Firstly, a validation ending case, Purdue flared cone, is chosen and the good agreements with measured data and DNS data are shown in \S\ref{sec:FlaredCone}; In \S\ref{sec:Mach4.5}, the flat plate at Mach 4.5 is used to investigate the importance of primary mode disturbance amplitude and the characteristics and the mechanism of the secondary instability; In addition, NPSE is employed to build the baseflow for another flat plate at Mach 6.0, and the corresponding spatial secondary instability analysis with different pressure gradients is presented in \S\ref{sec:Mach6.0} to suggest that the factor determining the saturated amplitude is the dispersion relation and evolution history of the primary disturbance wave; Finally, the conclusions are summarized, and topics of future investigations are discussed in \S\ref{sec:Conclusion}.


\section{Mathematical methods}\label{sec:mathematical}

Owing to the boundary layers investigated in this paper are all at a high speed, the compressible Navier-Stokes equations are introduced. Meanwhile, linear stability theory (LST), nonlinear parabolized stability equations (NPSE), secondary instability theory (SIT) are all solved for the present theoretical analysis as their high efficiency. It has been known that the compressible Navier-Stokes equations \citep{chang2003langley,chang2004lastrac}, LST \citep{1990Numerical} and NPSE \citep{bertolotti1991analysis,bertolotti1992linear,chang2003langley,chang2004lastrac,zhang2007verification,zhao2016improved} methods are described in much other work ; only a brief description of SIT is given in the following.


\iffalse

\subsection{Compressible Navier-Stokes equations for disturbances}

% The boundary layer equations of infinite swept wing, compressible linear stability theory equations, compressible parabolic stability equations and Bi-Global instability equations are solved for present theoretical analysis. Because these methods have matured, we only give a brief description for them in the following.
Generally, the 3D non-dimensional compressible Navier-Stokes equations have the form \citep{chang2003langley,chang2004lastrac}
\begin{subeqnarray}\label{eq:NS}
	\ \frac{\partial \rho}{\partial t}
	+ \frac{\partial (\rho v_j)}{\partial x_j} & = &
	0,\\[3pt]
	\rho \left(\frac{\partial v_i}{\partial t}
	+ v_j \frac{\partial v_i}{\partial x_j}\right) & = &
	%  -\frac{1}{\gamma M^2} \frac{\partial (\rho T)}{\partial x_i}%
	-\frac{\partial p}{\partial x_i}
	+\frac{1}{Re}\frac{\partial }{\partial x_j}\left[\mu(\frac{\partial v_i}{\partial x_j}
	+\frac{\partial v_j}{\partial x_i} -\frac{2}{3} \delta_{ij} \frac{\partial v_k}{\partial x_k})\right],\\[3pt]
	\rho \left(\frac{\partial T}{\partial t}
	+ v_j \frac{\partial T}{\partial x_j}\right) & = &
	\frac{1}{Re Pr} \frac{\partial }{\partial x_j} \left(\kappa \frac{\partial T}{\partial x_j}\right)
	+\frac{\gamma-1}{\gamma}\left(\frac{\partial p}{\partial t}+v_j\frac{\partial p}{\partial x_j}\right)
	\nonumber\\
	&& + \frac{(\gamma-1)M^2}{Re} \left[\mu\frac{\partial v_i}{\partial x_j}\left(\frac{\partial v_i}{\partial x_j}
	+\frac{\partial v_j}{\partial x_i}-\frac{2}{3} \delta_{ij} \frac{\partial v_k}{\partial x_k}\right)\right],
\end{subeqnarray}
where $t$ means the time, $x_j$ indicates the Cartesian coordinate, $\rho$ represents the density, $v_j$ denotes the velocity component. In addition, $T$ is the temperature, $\mu$ the dynamic viscosity, $\kappa$ the heat conductivity, $Pr$ the Prandtl number, $\delta_{ij}$ the Kronecker operator and $M$ the Mach number. In equation(\ref{eq:NS}), the coordinates are non-dimensionalized by a reference length $L^*_{ref}$, velocity by $U^*_{\infty}$, density by $\rho^*_{\infty}$, pressure by $\rho^*_{\infty} U^{*2}_{\infty}$, temperature by $T^*_{\infty}$, viscosity by $\mu^*_{\infty}$, heat conductivity by $\kappa^*_{\infty}$ and time by $L^*_{ref}/u^*_{\infty}$. Hence, the Reynolds number $Re$ is given by $Re=U^*_{\infty} L^*_{ref}/\nu^*_{\infty}$ where $\nu^*_{\infty}$ is the kinematic viscosity. In this paper, the perfect gas relation and the Sutherland formulation are employed. Furthermore, the frequency $\omega$ are non-dimensionalized by $2 \pi f L^*_{ref}/U^*_{\infty}$ where $f$ is the dimensional frequency in Hertz. Hereafter, the subscripts `0' and `$\infty$' represent the variables in the laminar mean flows and in the freestream, respectively.

In order to investigate the development of disturbances, the governing equations of disturbances must be derived. The instantaneous variable can be defined as the sum of perturbation $\phi'$ and base flow $\phi_0$, i.e.,
\begin{equation}\label{eq:disturbanceForm}
	\left. \begin{array}{ll}
		\displaystyle u'=u_0+u',v=v_0+v',w=w_0+w',p=p_0+p',\\[8pt]
		\displaystyle \rho=\rho_0+\rho',T=T_0+T',\mu=\mu_0+\mu',\kappa=\kappa_0+\kappa'.
	\end{array}\right\}
\end{equation}
Substituting equation(\ref{eq:disturbanceForm}) into the  Navier-Stokes equations and subtracting the control equations of the base flow, the compact scheme of perturbation equations can be obtained as follow
\begin{eqnarray}\label{eq:perturbationEquation}
	(\boldsymbol \Gamma \frac{\partial }{\partial t} &+& \boldsymbol A\frac{\partial }{\partial x}+ \boldsymbol B\frac{\partial }{\partial y}+ \boldsymbol C\frac{\partial }{\partial z}+ \boldsymbol D +\boldsymbol V_{xx} \frac{\partial^2 }{\partial x^2}+ \boldsymbol V_{yy}\frac{\partial^2 }{\partial y^2}+\boldsymbol V_{zz}\frac{\partial^2 }{\partial z^2} \nonumber\\
	&& +\boldsymbol V_{xy}\frac{\partial^2 }{\partial x \partial y} + \boldsymbol V_{yz}\frac{\partial^2 }{\partial y \partial z}+\boldsymbol V_{xz}\frac{\partial^2 }{\partial x \partial z} )\phi'= \boldsymbol F_n,
\end{eqnarray}
where $\phi'$ is defined as ($\rho',u',v',w',T'$) and the coefficient matrix $\boldsymbol \Gamma,\boldsymbol A,\boldsymbol B,\boldsymbol C,\boldsymbol D,\boldsymbol V_{xx},\boldsymbol V_{yy},$
$\boldsymbol V_{zz},\boldsymbol V_{xy},\boldsymbol V_{yz},\boldsymbol V_{xz}$ and nonlinear terms $\boldsymbol F_n$ with the curvatures can be derived and referred to the literature  \citep{chang2003langley,chang2004lastrac}.

\subsection{Linear stability theory}
Equations(\ref{eq:perturbationEquation}) are much complex and nonlinear, which can be simplified. A general simplified system for linear disturbances is the linear stability theory(LST)\cite{1990Numerical,Xu2020} . Because the wavelength of the disturbance is significantly shorter than the scale of base flow, the parallelism assumption can be adopted. The linear disturbance can be written as

\begin{equation}\label{eq:Disturbance_LST}
	\phi'(x,y,z,t)=\hat{\phi}(y) \mathrm{e}^{(\alpha x+\beta z -\omega t)} + \mathrm{c.c.},
\end{equation}
where $\phi'$ represents the five primitive disturbance variable \textit{$\rho',u',v',w',T'$} and $\hat{\phi}$ indicates the disturbance shape function for each disturbance variable. $\alpha, \beta$ and $\omega$ are the streamwise wavenumber, the spanwise wavenumber and the frequency, respectively.

With the parallel flow assumption and the disturbance form in equation(\ref{eq:Disturbance_LST}), retaining the derivatives in wall normal direction and removing the nonlinear terms in equation(\ref{eq:perturbationEquation}),
linear stability theory (LST) equations can be obtained as

\begin{equation}\label{eq:Equation_LST}
	\boldmath{L}\unboldmath \hat \phi=0,
\end{equation}
% The related $e^N$ method have been developed to maturity %\cite{}.
where $\boldmath{L}\unboldmath$ donates the linear operator of the LST and it is a function of $\alpha, \beta$ and $\omega$. Obviously, it is an eigenvalue problem with the homogeneous boundary conditions and describes the dispersion relation of the disturbance. This eigenvalue problem can be discredited by the five-point fourth-order central difference scheme in the wall-normalwise direction.

\subsection{Non-linear parabolized stability equations}
In order to simulate the linear and non-linear development of Mack mode disturbances in hypersonic boundary layers, the non-linear parabolized stability equations proposed by Bertolotti et al.\cite{bertolotti1991analysis,bertolotti1992linear} and enhanced for compressible flows\cite{chang2004lastrac}, are chosen here. In the NPSE method, non-parallel characteristic of the base flow and non-linearity of the disturbances are both taken into account for a more accuracy prediction. Compared with DNS, the NPSE method has almost the same accuracy with DNS method before transition, but costs less computing resources.

In the NPSE, the disturbance variables $\phi'_{NPSE}$ and non-linear term $\boldsymbol F_n$ take the form
\begin{equation}
	\label{eq:Disturbance_NPSE}
	\phi'_{NPSE}(x,y,z,t)=\sum_{m=-M}^M\sum_{n=-N}^N\widehat{\phi}_{mn}(x,y) e^{i\int \alpha_{mn}(\overline{x})d\overline{x} }e^{i(n\beta z -m\omega t)} .
\end{equation}
\begin{equation}
	\label{eq:Disturbance_Fn}
	\boldsymbol F_n(x,y,z,t)=\sum_{m=-M}^M\sum_{n=-N}^N\hat {\boldsymbol F}_{mn} e^{i(n\beta z -m\omega t)}
\end{equation}
\quad Substituting the sum of base-flow variables and disturbances with the form in Eq.(\ref{eq:Disturbance_NPSE}), the compact scheme of non-linear parabolized stability equation can be obtained by eliminating the ellipticity
\begin{equation}
	\label{eq:NPSE}
	(\hat {\boldsymbol A}_{mn} \frac{\partial }{\partial x}+\hat {\boldsymbol B}_{mn}\frac{\partial}{\partial y}+\hat {\boldsymbol V}_{mn}\frac{\partial^2}{\partial y^2}+ \hat {\boldsymbol D}_{mn})\widehat{\phi}_{mn}=\hat {\boldsymbol F}_{mn}
\end{equation}
subject to the following boundary conditions
\begin{subeqnarray}\label{bc:NPSE}
	&\hat u_{mn}=\hat v_{mn}=\hat w_{mn}=\hat T_{mn}=0\qquad y=0;\\[1mm]
	&m=n=0,\qquad \hat\rho_{mn}=\hat u_{mn}=\hat w_{mn}=\hat T_{mn}=0,\quad \partial \hat v_{mn} /\partial y =0\qquad y\to y_\infty\\[1mm]
	&\mathrm{else},\qquad \hat\rho_{mn}=\hat u_{mn}=\hat v_{mn}=\hat w_{mn}=\hat T_{mn}=0,\qquad y\to y_\infty
\end{subeqnarray}
where the detailed expression of matrixes $\hat {\boldsymbol A}_{mn},\hat {\boldsymbol B}_{mn},\hat {\boldsymbol V}_{mn},\hat {\boldsymbol D}_{mn}$ and the non-linear term $\hat {\boldsymbol F}_{mn}$ can be found in Refs. \cite{chang2003langley,chang2004lastrac} and \cite{zhang2007verification}. Note that the under-relaxation method  \cite{chang2004lastrac,zhao2016improved} is selected in this paper for a better convergent solution. The PSE method is solved with the finite difference method which includes the five-point forth-order central difference scheme in the normal direction and backward difference scheme of second order accuracy in the streamwise direction. Note that the spectrum method is employed in the periodic spanwise and temporal directions.
\fi


\subsection{Secondary Instability Theory}




The Floquet-based secondary instability theory (SIT) has been widely applied to analyse the secondary instability mechanism \cite{herbert1988secondary,hogberg1998secondary,malik1999secondary,theofilis2011global,ren2015secondary,chen2017interactions,xu2019secondary}. It describes the instability characteristics in boundary layers when the primary mode disturbance grows up to a sufficiently large  amplitude. The composition of the primary mode and the 2D mean-flow can be seen as the new base-flow for secondary instabilities. In other words, the new base-flow can be written as $\bm{U}_b(x,y,t)=\bm{\phi}_0(y)+\sum\limits_{m=-M}^M\widehat{\bm{\phi}}_{m}(x,y) e^{i\int \alpha_{m}(\overline{x})d\overline{x} }e^{-i m\omega t}$ based on the NPSE results, where $\bm{\phi}_0$ is the original 2D laminar mean flow solution, $\widehat{\bm{\phi}}_m$ is the primary mode and its harmonic modes. Obviously, the base-flow is a function of time $t$ in the original frame ($x, t$). To eliminate the time $t$, Galilean transformation with the form of $\widetilde{x}=x-c_r t$ is introduced where $c_r$ stands for the phase velocity of the primary Mack mode disturbance. It means that the new base-flow in the new frame is stationary so that the Floquet theory can be easily adopted. It is obviously that the high order disturbances can be governed by the dynamics system
\begin{equation}
	\label{eq:Equation_BiGlobal}
	\boldmath{\widetilde{\bm{L}}}\unboldmath \bm{\phi}'_S=0
\end{equation}
where $\bm{\phi}'_S$ is the secondary instability disturbance corresponding to the five primitive variables and the operator $\boldmath{\widetilde{\bm{L}}}\unboldmath$ is the function of the new base-flow $\widetilde{\bm{U}}_b(\widetilde x,y)$. Due to the local growth rate primary mode disturbance is significantly less than the one of the secondary instability disturbance when the primary mode grows to a sufficiently large amplitude, a quasi-parallel assumption can be adopted for the insignificant variation of the primary mode amplitude. Moreover, it is seen that the secondary instability is periodic in the streamwise direction $\widetilde{x}$.

In Herbert's paper, the general form of the solution can be expressed as
\begin{equation}
 \label{eq:Equation_BiGlobal_GeneralForm}
 \phi_S(\widetilde x, y, z, t)=e^{\gamma \widetilde x}e^{\sigma t} e^{i\beta_2 z} \sum_j \widetilde \phi_{S, j}(y) e^{i (j \alpha \widetilde x)}.
\end{equation}

Here, there are four real quantities $\sigma_r, \sigma_i, \gamma_r, \gamma_i$ associated with the secondary instability. $\beta_2$ stands for the wave number in $z$ direction. It is noticed that
\begin{equation}
e^{\sigma t}e^{\gamma \widetilde x}=e^{(\sigma-\gamma c_r)t}e^{\gamma x}.
\end{equation}

In a temporal mode, $\gamma_r=0$ and the growth rate is represented by $\sigma_r$, while $\sigma_i-\gamma_i c_r t$ is the frequency shift
with respect to the primary mode disturbance in the original frame $(x, t)$ and $\gamma_i$ is the streamwise wave number shift. $\gamma_i=\epsilon \alpha$ measures the detuning of the phase difference between the new base-flow $\widetilde{\bm{U}}_b(\widetilde x,y)$ and the secondary disturbance with the Floquet parameter $\epsilon \in [0,0.5]$. When $\epsilon=0$, the frequency shift and streamwise wave number shift are both zero, i.e. it is a fundamental mode;
when $\epsilon=0.5$, the streamwise wave number shift is $0.5\alpha$ and the frequency shift is $-0.5\omega$, i.e. it is a subharmonic mode; when $ 0 \textless \epsilon \textless 0.5$,
it is a detuned mode. As a result, the temporal mode disturbance can be expressed as
$\phi_S(\widetilde x, y, z, t)=e^{i\epsilon\alpha \tilde x}e^{\sigma t} e^{i\beta_2 z}\sum_j \widetilde \phi_{S, j}(y) e^{i (j \alpha \widetilde x)}$. This expression imjplies that the detuning of the frequency and the streamwise wave number are both given by $\epsilon$ to solve $\sigma$ and the streamwise wave number shift excluding the detuning is zero. 

In a spatial mode, $\sigma_r-\gamma_r c_r=0$ and $\gamma_r$ governs the spatial growth of the disturbance. $\gamma_i$ is the shift in the streamwise wave number and the detuning of the frequency
is $\sigma_i-\gamma_i c_r$. In order to solve the spatial problem, the detuning also should be given to determine the actual streamwise wave length. However, the streamwise wave number shift excluding the detuning, which is an unknown quantity, is difficult to define. It will take difficulty to write a discrete form to a general eigenvalue problem $A\varphi=\lambda B\varphi$ for a general FORTRAN code to both the temporal mode and the spatial mode secondary instability.

As a result, we define the secondary instability disturbance a similar but different expression as:
\begin{equation}
\phi_S\left(\widetilde{x},y,z,t\right)=e^{\sigma\ t}e^{\gamma\widetilde{x}}e^{i\epsilon\alpha\widetilde{x}}e^{i\beta_2z}\sum_{j}{\widetilde{\phi}_{S,j}\left(y\right)e^{i\left(j\alpha\widetilde{x}\right)}}
\end{equation}


In this form, $\gamma_i$ is different from its definition in Herbert’s form. Because $e^{\sigma t}e^{\gamma\widetilde{x}}=e^{\left(\sigma-\gamma c_r\right)t}e^{\gamma x}$, $\sigma_r-\gamma_rc_r$  and $ \gamma_r$ are still the temporal and spatial growth rate respectively. $\sigma_i-\gamma_ic_r$ now is the frequency shift excluding the detuning and $\gamma_i$ is the streamwise wave number shift excluding the detuning. In temporal mode, $\gamma_r=\gamma_i=0$ means that the spatial growth rate and the streamwise wave number shift excluding the detuning are both zero. The parameter $\epsilon$ is still adopted to define the detuning. As a result, $\sigma_i$ is still the frequency shift excluding the detuning as that in the Herbert’s expression and $\sigma_r$ is the temporal growth rate. In the spatial mode, the detuning parameter $\epsilon$ is given. In addition, the temporal growth rate $\sigma_r-\gamma_r c_r=0$ and the frequency shift excluding the detuning $\sigma_i-\gamma_ic_r=0$ which is given in the spatial mode to determine the spatial status. In other word, $\sigma=\gamma c_r$ and $\gamma$ is the unknown eigenvalue value to determine the spatial growth rate and the streamwise wave number shift excluding the detuning.\ As a result, the secondary instability disturbance in the spatial mode here can be simplified as:
\begin{eqnarray}
	\phi_S\left(\widetilde{x},y,z,t\right)&=&{e^{\gamma Crt}e}^{\gamma\widetilde{x}}e^{i\epsilon\alpha\widetilde{x}}e^{i\beta_2z}\sum_{j}{\widetilde{\phi}_{S,j}\left(y\right)e^{i\left(j\alpha\widetilde{x}\right)}} \nonumber\\
    &&=e^{\gamma\left( \widetilde{x}+c_rt\right)e^{i\epsilon\alpha\widetilde{x}}e^{i\beta_2z}}\sum_{j}{\widetilde{\phi}_{S,j}\left(y\right)e^{i\left(j\alpha\widetilde{x}\right)}}\nonumber\\
    &&=e^{\gamma x}e^{i\epsilon\alpha\widetilde{x}}e^{i\beta z}\sum_{j}{\widetilde{\phi}_{S,j}\left(y\right)e^{i\left(j\alpha\widetilde{x}\right)}}.
\end{eqnarray}
In this case, $\gamma$ is the unknown eigenvalue to be solved by an eigenvalue problem. When the shift $\gamma_i=0$, it means that the secondary instability disturbance is “phase locked” to the primary mode disturbance. As a result, if the SIA mode is subharmonic ($\epsilon=0.5$), the frequency of the subharmonic mode is $0.5\omega$ and the streamwise wave number is $0.5\alpha$; if the SIA is fundamental ($\epsilon=0$), the frequency of the fundamental mode is $\omega$ and the streamwise wave number is $\alpha$.

 Obviously, this is a complex eigenvalue problem that reads
\begin{equation}
	\label{eq:Equation_BiGlobal_hat}
	\boldmath{\widehat{\bm{L}}}\unboldmath \widetilde{\bm{\phi}}_{S}=0
\end{equation}
subject to the the homogeneous boundary conditions. The forth-order central difference scheme and the Fourier spectral method are used for the numerical discretization in the wall-normal direction and
the streamwise direction, respectively.

% \quad Note that $x_2,y_2,z_2$ are vortex axis coordinates in vortex axis direction, vortex axis coordinates normal to the wall and vortex axis coordinates perpendicular to the $x_2-y_2$ plane.



\subsection{Two-dimensional similarity flow}\label{sec:FS}
% It has been known that the the pressure gradient has a significant effect on the growth of the primary mode disturbance. As a result, it makes the saturated amplitude of the primary perturbance different.
In this paper, the pressure gradient effect on a secondary instability is investigated. For investigating this effect, the Falkner-Skan similarity solution is chosen as the basic flow. In the following, the equations of these basic flow are introduced.

In order to localize the non-local variables, two-dimensional compressible similarity equations are introduced to build the relation functions.
\begin{equation}
	\left. \begin{array}{ll}
		\displaystyle \xi =\int_0^{\infty} \rho_{\infty}^* U_{\infty}^* \mu_{\infty}^* d x^*\,\\[8pt]
		\displaystyle \eta  =\frac{U_{\infty}}{\sqrt{2 \xi}}\int_0^{y^*} \rho \,d y^*=\sqrt{\frac{\rho_{\infty}^* U_{\infty}^*}{2 \mu_{\infty}^* x^*}}\int_0^{y^*} \frac{\rho^*}{\rho_{\infty}^*}\, d y^*,
	\end{array}\right\}
	\label{symbc}
\end{equation}

With Illingworth transformation, the two-dimensional boundary equations can be written as
\begin{equation}
	\label{eq:FS1}
	\left( \frac{\rho \mu}{\rho_e \mu_e} f''     \right)' +f f''+ \beta_H \left(\frac{\rho_e}{\rho} -f'^2 \right),
\end{equation}
\begin{equation}
	\label{eq:FS2}
	\left( \frac{1}{Pr} \frac{\rho \kappa}{\rho_e \kappa_e} f''     \right)' +f g'+ (\gamma_H-1) M_e^2 \beta_H \frac{\rho \mu}{\rho_e \mu_e} +(\gamma_H-1)M_e^2 \beta_H f'^2 (f'^2 -g),
\end{equation}
with the boudary conditions
\begin{equation}
	\left. \begin{array}{ll}
		\displaystyle f=f'=0, g=g_w, (or\quad g'=0 \quad for\quad adiabatic \quad wall)\quad \eta=0,\\[8pt]
		\displaystyle f'=g=1, \quad \quad \quad \quad \eta=\infty.
	\end{array}\right\}
	\label{symbc}
\end{equation}

In the equations above, $f'=u/U_e$  and  $g=T/T_e$ indicate the velocity profile and temperature profile, respectively.  $\beta_H = \frac{2 \xi}{U_e} \frac{d U_e}{d \xi}$ is the Falkner-Skan pressure gradient parameter and $\gamma_H$ the ratio of specific heats. Note that the subscript `$e$' means the variables at the edge of boundary layer and the primes stand for derivatives with respect to $\eta$. When $\beta_H$ equals zero, the equation(\ref{eq:FS1}) and equation(\ref{eq:FS2}) become the Blasius similarity equations. All the coordinates are non-dimensionalized by the displacement thickness $\delta_0^*$ at the position of $x^*=x_0^*$. The definition of displacement thickness is
\begin{equation}
	\delta^*=\int_0^\infty \left(1-\frac{\rho^* u^*}{\rho_{\infty}^* U_{\infty}^*} \right)\, d y^*  = \int_0^\infty \left[g(\eta) -f'(\eta)\right] \sqrt{\frac{2 \mu_{\infty}^* x^*}{\rho_{\infty}^* U_{\infty}^*}} \, d \eta,
\end{equation}
yielding the non-dimensional coordinate in the normal direction
\begin{equation}\label{eq:flatplate_y}
	y=\frac{y^*}{\delta^*}=\sqrt{\frac{x}{x_0}}\frac{\int_0^{\eta}g(\eta)\,d \eta}{ \int_0^\infty \left[g(\eta) -f'(\eta)\right] \, d \eta}.
\end{equation}

Then, the Reynolds number is defined as $Re_\delta =\rho_{\infty}^* U_{\infty}^* \delta^* / \mu_{\infty}^*$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Code validation}
\begin{figure}
	\centering
	\includegraphics[width=70mm]{photos/NPSE_DNS_com.eps}% Here is how to import EPS art
	\caption{Comparisons of the NPSE results and the DNS data in the case from Ref. \cite{2014Nonlinear_yumin}.}\label{fig:NPSE_DNS_com}
\end{figure}
Firstly, the NPSE method is validated using the published DNS case \cite{2014Nonlinear_yumin}. In a Mach 6 flat-plate boundary layer, the 2D primary Mack mode disturbance and a pair of 3D fundamental mode disturbances are all placed at the inlet position ($x=450$). They have the initial amplitude of 1$\times 10^{-2}$ and 1$\times 10^{-4}$, respectively. The freestream and wall temperature are 250.35K and 1000K, respectively. The unit Reynolds number is 4.75027$\times 10^5/$m. DNS is employed to investigated the non-linear interaction mechanism.  As shown in Fig. \ref{fig:NPSE_DNS_com}, a good agreement of the amplitude evolution between the NPSE results and DNS data is obtained, which proves that the NPSE method is trustworthy for the problem studied in this Paper.
\begin{figure}

	\centering

	\subfigure[Subharmonic instabilities based on a first mode (2-D) primary at	$M_\infty$ = 4.5]{
		\begin{minipage}{60mm}
			\centering
			\includegraphics[width=60mm]{photos/V1.eps}
		\end{minipage}
	}
	\subfigure[Subharmonic instabilities based on a Mack mode primary at $M_\infty$ = 4.5]{
		\begin{minipage}{60mm}
			\centering
			\includegraphics[width=60mm]{photos/V2.eps}
		\end{minipage}
	}
	\subfigure[Fundamental instabilities based on a first mode (2-D) primary at	$M_\infty$ = 1.6]{
		\begin{minipage}{60mm}
			\centering
			\includegraphics[width=60mm]{photos/V3.eps}
		\end{minipage}
	}
	\caption{The comparisons of the secondary instabilities for Mach 4.5 and Mach 1.6 flat plate boundary layers between the present results and the reference data from Ref.\cite{ng1992secondary}.}\label{fig:validation}
\end{figure}

Secondly, three temporal growth-rate calculation cases with various primary amplitudes in supersonic flat plate boundary layers, including the subharmonic instabilities based on a first mode (2-D) primary at	$M_\infty$ = 4.5, the subharmonic instabilities based on a Mack mode primary at $M_\infty$ = 4.5 and the fundamental instabilities based on a first mode (2-D) primary at	$M_\infty$ = 1.6 from the Reference \cite{ng1992secondary,li2012secondary} are chosen for the validations of the present SIT code. As shown in Fig. \ref{fig:validation}, the results of the present SIT code agree well with the reference data, which indicates that the present SIT code is reliable.




%In order to investigate the mechanism of the fundamental mode of the primary Mack mode disturbance in a hypersonic boundary layer, the boundary layer on the flared cone in BAM6QT and the two-dimensional Falkner-Skan compressible boundary layer are selected. The former one has the experimental and direct numerical results for the comparison, while the latter one has a simple geometric configuration. Furthermore, selecting a flat plate is more conducive to verify some conjectures.
% \subsection{Energy balance analysis}

\section{Results and discussions}\label{sec:TestCases}
% In this section, the two-dimensional Falkner-Skan compressible boundary layer similarity equations are introduced.




\subsection{Secondary instability in Ma=4.5 flat-plate flow}\label{sec:Mach4.5}
In the previous section, the codes for NPSE and SIT have been validated and confirmed. However, when we conducted the secondary instability analysis on the Mach 4.5 flat plate flow \citep{ng1992secondary}, some problems were encountered. 

The case for secondary instability analysis with $\mathrm{Re}= 10,000$ $\alpha=2.52$ were employed here. \ref{fig:M4p5_GR_Amp} plots the continuous solutions of the spatial/temporal growth rate $\sigma_r$ and the corresponding phase $\sigma_i,\gamma_i$ of secondary instability versus the primary amplitude $A$. When $\beta_2$=2.1, the growth-rate of the subharmonic mode is the largest. It can be seen that the phases almost equal to zero, which satisfies the ``phase-locked'' characteristic of a secondary instability \citep{wu1996weakly,wu1996interaction,wu2007catalytic,wu2019nonlinear}. What is more, as the amplitude $A$ of primary Mack mode increases from 6\% to 100\%, the growth rate of the fundamental family grows up rapidly and finally surpasses the subharmonic family \citep{hader2016laminar,hader2017fundamental,xu2020secondary,xu2021pressureg}.  In a word, the fundamental resonance dominates the secondary instability when the primary amplitude exceeds a threshold value. 

In the traditional way, when the temporal growth-rate is solved, the spatial growth-rate can be obtained through the well-known Gaster transformation. This classical method has been employed and the results are displayed in figure \ref{fig:M4p5_GR_Amp_phase}. From the figure, in subharmonic resonance, the growth-rate of the temporal mode after Gaster transformation is in good agreement with the growth rate of the exact spatial mode, which is in line with traditional cognition. However, in the fundamental resonance, the growth-rate of the temporal mode after Gaster transformation is very different from the growth rate of the exact spatial mode, especially when the primary amplitude $A$ is large. This is not in line with the traditional cognition. 

\begin{figure}
	\psfrag{A}[][]{$A$}
	\psfrag{M}[][]{$(a)$} \psfrag{N}[][]{$(b)$}
	\psfrag{G}{$\sigma_r,\gamma_r$} \psfrag{P}{$\sigma_i,\gamma_i$}
	
	\centerline{\includegraphics[scale=0.6]{photos/M4p5_GR_Amp.eps}}% Images in 100% size
	\caption{Secondary instability growth rate against the primary amplitude of Mack mode disturbances through the spatial SIT and temporal SIT when $\beta_2=2.1$.(\textit{a}) Growth rate. and (\textit{b}) phase.}
	\label{fig:M4p5_GR_Amp}
\end{figure}


\begin{figure}
	\psfrag{A}[][]{$A$}
	\psfrag{M}[][]{$(a)$} \psfrag{N}[][]{$(b)$}
	\psfrag{G}{$\sigma_r,\gamma_r$} \psfrag{P}{$\sigma_i,\gamma_i$}
	\centerline{\includegraphics[scale=0.6]{photos/M4p5_GR_Amp_phase.eps}}% Images in 100% size
	\caption{Comparison of the secondary instability growth rate against the primary amplitude of Mack mode disturbances through the spatial SIT and Gaster transformation when $\beta_2=2.1$. (\textit{a}) fundamental resonance. and (\textit{b}) subharmonic resonance.}
	\label{fig:M4p5_GR_Amp_phase}
\end{figure}

Why does this phenomenon occur, and what are the limitations of the traditional Gaster transformation? Below we will explain when this traditional knowledge cannot be used from the perspective of mathematics.

\subsection{Gaster transformation}
Original derivation method (Gaster, 1962, a note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability)

The Gaster transformation can describe the conversion relationship between the temporal mode and the spatial mode in the flow instability problem. The specific derivation process is:
\subsubsection{Derivation method 1}
Assuming that the relationship between the wave number and frequency of the disturbance is an analytic function, it can satisfy the Cauchy-Riemann relationship:

\begin{equation}
	\frac{\partial \omega_r}{\partial \alpha_r}=\frac{\partial \omega_i}{\partial \alpha_i},\qquad
	\frac{\partial \omega_i}{\partial \alpha_r}=\frac{\partial \omega_r}{\partial \alpha_i}
\end{equation}
So we integrate $\alpha$ from temporal mode (T) to spatial mode (S) on both sides of the above formula, we have

\begin{equation}
	\omega_i(S)-\omega_i(T)=-\omega_i(T)=\int_0^{\alpha_i(S)}\frac{\partial\omega_r}{\partial\alpha_r} \rm{d}\alpha_i 
	\\
	\omega_r(S)-\omega_r(T)=-\int_0^{\alpha_i(S)}\frac{\partial\omega_i}{\partial\alpha_r} d\alpha_i
\end{equation}
It is believed that $\alpha_r$ remains unchanged from temporal mode (T) to spatial mode (S), that is, $\alpha_r(S)=\alpha_r(T)$.

Usually we think that $\omega_r\gg \omega_i$ and $\alpha_r \gg \alpha_i$ exist for both complex frequency and complex wavenumber, so that we can get $\frac{\partial\omega_i}{\partial\alpha_r}=O (\omega_{im})$. Here, the subscript $m$ is marked as the maximum value of the entire change process under a given span wave number and Reynolds number (that is, given other conditions). Similarly, $\frac{\partial\alpha_i}{\partial\omega_r}=O(\omega_{im})$ must exist. In this way, the above derivation can be further simplified as:
\begin{equation}
	\omega_r(S)=\omega_r(T)+O(\omega_{im}\alpha_i(S))
\end{equation}

If the small amount is ignored, it gets $\omega_r(S)=\omega_r(T)$, which is the conversion relationship between wave numbers.

For the growth rate, we can first perform Taylor expansion on any point in $\partial\omega_r/\partial\alpha_r$ from 0 to $\alpha_i(S)$, then we have:

\begin{equation}
	\frac{\partial\omega_r}{\partial\alpha_r}=\frac{\partial\omega_r}{\partial\alpha_r}|_{\alpha_i^*}+\frac{\partial^2\omega_r}{\partial\alpha_r\partial\alpha_i}|_{\alpha_i^*}(\alpha_i-\alpha_i^*)+...
\end{equation}

Substituting the above expansion into the derivation relation given above, there are:
\begin{equation}
	\omega_i(T)=-\alpha_i(T)\frac{\partial\omega_r}{\partial\alpha_r}|_{\alpha_i^*}-\{0.5\alpha_i^2(S)-\alpha_i(S)\alpha_i^*\}\frac{\partial^2\omega_r}{\partial\alpha_r\partial\alpha_i}|_{\alpha_i^*}+...
\end{equation}
Substituted into the Cauchy relationship, it can be further written as:
\begin{equation}
	\omega_i(T)=-\frac{\partial\omega_r}{\partial\alpha_r}|_{\alpha_i^*}+O(\omega_{im}\alpha_i(S))
\end{equation}
Also ignore the small amount, the spatio-temporal relationship between the growth rates can be obtain
\begin{equation}
	\frac{\omega_i(T)}{\alpha_i(S)}=-\frac{\partial\omega_r}{\partial\alpha_r}
\end{equation}
Here, $\partial\omega_r/\partial\alpha_r$ can use the value of any point in the time mode and the space mode.

From the above derivation process, we can see that the premise of Gaster transformation is to discard the terms of $O(\alpha_i(S)\omega_{im})$, and the premise is that the real part of the complex wave number and frequency is much larger than the imaginary part.

\subsubsection{Derivation method 2}
The Gaster transformation can actually be derived directly according to the nature of the implicit function using the full differential, and introduce similar assumptions as above, and finally get the same transformation form.

We consider that the dispersion relationship satisfies the following relationship
\begin{equation}
	D(\omega, \alpha, Re)=0
\end{equation}
Then there is the following total differential relationship:
\begin{equation}
	\rm{d}D(\omega, \alpha, Re)=\frac{\partial D}{\partial \omega}d\omega+\frac{\partial D}{\partial \alpha}d\alpha+\frac{\partial D}{\partial Re}dRe=0
\end{equation}
Therefore, there is the following relationship:
\begin{equation}
	d\omega|_{\alpha_0}=-dRe\frac{\partial D}{\partial Re}/\frac{\partial D}{\partial \omega}
	\\
	d\alpha|_{\omega_0}=-dRe\frac{\partial D}{\partial Re}/\frac{\partial D}{\partial \alpha}
	\\
	d\omega|_{Re_0}=-d\alpha\frac{\partial D}{\partial \alpha}/\frac{\partial D}{\partial \omega}=\frac{\partial \omega}{\partial \alpha}|_{Re_0}d\alpha
\end{equation}
Therefore, we have
\begin{equation}
	d\omega|_{\alpha_0}=d\alpha|_{\omega_0}\frac{\partial D/\partial \alpha}{\partial D/\partial \omega}=(-\frac{\partial \omega}{\partial \alpha}|_{Re_0})d\alpha|_{\omega_0}=-C_g d\alpha|_{\omega_0}
\end{equation}

Integrating $\alpha$ from temporal mode (T) to spatial mode (S) at the same time on both sides of the above formula, then:
\begin{equation}
	\int_{\alpha(T)}^{\alpha(S)}d\omega=\int_{\alpha(T)}^{\alpha(S)}-\frac{\partial \omega}{\partial \alpha}d\alpha
\end{equation}

Integrating the above equations in the complex plane can also get a similar conclusion as the previous derivation. In this derivation process, it is still necessary to discard the terms of $O(\alpha_i(S)\omega_{im})$, but the premise is that the real part of the complex wave number and frequency is much larger than the imaginary part, or the real and imaginary parts The difference is at least an order of magnitude.
\subsection{Gaster Transformation in the Problem of Secondary Instability}
For the problem of secondary instability, the Gaster transformation problem still exists, and we can derive the Gaster transformation in the secondary instability problem based on the similar relationship above. From the perspective of the classification of secondary instability, it can be divided into secondary instability of streaks and Herbert's classic secondary instability.
\subsubsection{Secondary instability of streaks}
In the secondary instability of streak, considering the three-dimensionality of the basic flow $U(y, z)$ along the span, based on the Floquet theory, the instability disturbance can be written as:
\begin{equation}
	\phi^\prime(x, y, z, t)=\sum_{m=-\infty}^{\infty}\hat \phi(y)_m e^{im\beta z}e^{i\epsilon \beta z}e^{i(\alpha x-\omega t)}
\end{equation}
Among them, $\beta$ corresponds to the spread wave number of the elementary flow.
According to the derivation of the aforementioned Gaster transformation, it can still give:
\begin{equation}
	d\omega|_{\alpha_0}=d\alpha|_{\omega_0}\frac{\partial D/\partial \alpha}{\partial D/\partial \omega}=(-\frac{\partial \omega}{\partial \alpha}|_{Re_0})d\alpha|_{\omega_0}=-C_g d\alpha|_{\omega_0}
\end{equation}
At this point, we can notice that for Streak's secondary instability problem, whether it is a space mode or a time mode, the eigenvalues have the feature that the real part is much larger than the imaginary part, which means that the Gaster transformation is effective:
\begin{equation}
	\omega_r(T)=\omega_r(S)\\
	\omega_i(T)=-\alpha_i(S)\frac{\partial\omega_r}{\partial\alpha_r}
\end{equation}


\subsubsection{Herbert's secondary instability problem}
For Herbert's secondary instability problem, the basic flow is $U(x, y, t)=U_b(y)+(A\hat u(y)exp(i(\alpha x- \omega t))+c.c.) $. Galileo transformation $x^\prime=x-c_r t$ is usually introduced, and $t$ is eliminated to make the elementary stream $U(x', y)$. Here $c_r=\omega_r/\alpha_r$ corresponds to the phase velocity of the first instability disturbance, and $\omega_r$ and $\alpha_r$ actually correspond to the frequency and flow direction wave number of the first instability disturbance. At this time, $U(x', y)$ can be regarded as the basic flow of the secondary instability problem at this time, so the instability disturbance can be written as:

\begin{equation}
	\phi^\prime(x', y, z, t)=e^{\sigma t}e^{\gamma x'}e^{i\beta_2 z}e^{i\epsilon \alpha x'}\sum_{m=-\infty}^{\infty}\hat \phi(y)_m e^{im\alpha x'}
\end{equation}

In this definition, $\sigma_r$ and $\gamma_r$ correspond to the time and space growth rates of the disturbance, while the imaginary parts $\sigma_i$ and $\gamma_i$ are related to the phase shift of the disturbance.

Note that 
\begin{equation}
	e^{\sigma t}e^{\gamma x'}=e^{(\sigma-\gamma C_r)t}e^{\gamma x}
\end{equation}

According to the above formula, it can be known that the real part of $\sigma-\gamma C_r$ corresponds to the time growth rate of the disturbance, and the imaginary part corresponds to the movement of the frequency except Detuning; in the same way, the real part of $\gamma$ corresponds to the disturbance The spatial growth rate of, the imaginary part corresponds to the movement of the flow wave number outside of Detuning. Usually for the fundamental frequency or subharmonic mode, the perturbation has a phase-locked property, that is, the phase velocity of the mode is the same as the phase velocity of the first instability perturbation, so it corresponds to the time mode, $\gamma_i=\gamma_r=0$; corresponding For spatial mode, there is $\sigma-\gamma C_r=0$. In this way, the perturbation of the secondary instability under different span wave numbers beta can be solved.

Similarly, we can imitate the previous derivation and give the relationship between time and space, that is, existence:
\begin{equation}
	d\sigma|_{\gamma_0}=d\sigma|_{\gamma_0}\frac{\partial D/\partial \gamma}{\partial D/\partial \sigma}=(-\frac{\partial \sigma}{\partial \gamma}|_{Re_0})d\gamma|_{\sigma_0}
\end{equation}

However, we can find at this time that since the frequency shift in Herbert's secondary instability is close to 0, in fact, the difference between the real and imaginary parts of $\sigma$ and $\gamma$ is very small. The conditions of $\sigma_r \ll \sigma_i$ and $\gamma_r \ll \gamma_i$ are not satisfied, which makes it impossible for us to ignore the high-order small quantities of the growth rate as in the previous Gaster transformation derivation, or in other words, the aforementioned The quantity omitted in the derivation cannot be ignored at this time. This means that it is necessary to strictly calculate the integral of the relationship in the complex plane to obtain the time-space correspondence. However, the purpose of using time-space transformation is to  transform the results obtained from the time mode into the space mode easily, thereby reducing the difficulty and the amount of calculation for solving the space mode. If you need to calculate complicated complex plane integrals, you can get the corresponding Relationship, then obviously this transformation relationship loses its meaning at this time.

Obviously, due to the transformation relationship introduced directly according to the eigenvalues $\sigma$ and $\gamma$, the real and imaginary parts are both small, making the transformation process impossible to simplify, so we can use another form to give the temporal-spatial transformation relationship.

We noticed that the flow direction wave number and frequency of the disturbance, except for the parts of $\gamma$ and $\sigma$, the other parts still contribute to the flow direction wave number and frequency. Therefore, we can restore the expression of the disturbance in the $x$ coordinates. Write and organize, then:
\begin{eqnarray}
\phi^\prime(x, y, z, t) &=& e^{(\sigma_r-\gamma_rC_r)t}e^{\gamma_r x}e^{i\beta_2 z}\sum_{m=-\infty}^{\infty}\hat \phi(y)_m e^{i\theta}
	\theta \nonumber\\
	&=& (m+\epsilon)\alpha x-(m+\epsilon)\omega t+\gamma_i x+(\sigma_i-C_r\gamma_i)t\nonumber\\
	&=& (m+\epsilon+\frac{\gamma_i}{\alpha})\alpha x-(m+\epsilon-\frac{\sigma_i-C_r\gamma_i}{\omega})\omega t.
\end{eqnarray}

At this time we set
\begin{equation}
	\tilde \sigma=-(\sigma_r-\gamma_r c_r)+i(m+\epsilon-\frac{\sigma_i-C_r\gamma_i}{\omega})\omega
	\\
	\tilde \gamma=\gamma_r+i(m+\epsilon+\frac{\gamma_i}{\alpha})\alpha
\end{equation}

Obviously, the dispersion relationship is now $\tilde D(\tilde\sigma, \tilde\gamma)=0$. According to the previous derivation of the time-space relationship, we can get:
\begin{equation}
	d\tilde\sigma|_{\tilde\gamma_0}=d\tilde\sigma|_{\tilde\gamma_0}\frac{\partial D/\partial \tilde\gamma}{\partial D/\partial \tilde\sigma}=(-\frac{\partial \tilde\sigma}{\partial \tilde\gamma}|_{Re_0})d\tilde\gamma|_{\tilde\sigma_0}
\end{equation}
We can notice that since $\sigma_i \ll \omega$ and $\gamma_i \ll \alpha$ and $\sigma_r=O(10^{-2})$ and $\gamma_r=O(10^{-2})$,
	The difference between the real and imaginary parts of the redefined $\tilde \sigma$ and $\tilde \gamma$ is at least one order of magnitude, so the Gaster transformation derivation process described above can be continued. According to the derivation of the Gaster transformation, ignoring a small amount, we can get:
	
\begin{equation}
	\tilde\sigma_i(T)=\tilde\sigma_i(S)
	\\
	\tilde\sigma_r(T)=-\tilde\gamma_r(S)\frac{\partial \tilde\sigma_i}{\partial \tilde\gamma_i}
\end{equation}	


\subsubsection{Discussion on the relationship of temporal-spatial transformation in Herbert's secondary instability}	

According to the above derivation, considering the magnitude relationship, it is not difficult to find the following relationship between
\begin{equation}
	-\frac{\tilde\sigma_r(T)}{\tilde \gamma_r(S)}=\frac{\sigma_r}{\gamma_r}=\frac{\partial \tilde\sigma_i}{\partial \tilde\gamma_i}=\frac{\partial ((m+\epsilon)\omega)}{\partial((m+\epsilon)\alpha)}
\end{equation}

By definition, $\epsilon$ is a Floquet parameter, between 0 and 0.5, and $m$ is any integer.

When $\epsilon \gg 0$, obviously $m+\epsilon\neq 0$, at this time:
\begin{equation}
	\frac{\sigma_r}{\gamma_r}=\frac{\partial\omega}{\partial\alpha}
\end{equation}
Since for Herbert's secondary instability problem, both $\omega$ and $\alpha$ are given, and $\omega=c_r \alpha$, obviously:

\begin{equation}
	\frac{\sigma_r}{\gamma_r}=\frac{\partial(c_r\alpha)}{\partial\alpha}=c_r
\end{equation}

This explains why it is possible to directly use this formula to complete the time-space mode conversion in the subharmonic mode, that is, the conversion formula given in Herbert's paper.

When $\epsilon \rightarrow 0$ or $\epsilon=0$, for the component of $m \neq 0$, the above formula still holds. However, for the component with $m=0$, the value of the partial derivative cannot be given at this time. This makes the transformation relationship invalid, and it is impossible to simply give the transformation relationship. Therefore, only the secondary instability problem of the spatial model can be directly calculated to obtain the spatial growth rate.

In fact, when the partial derivative value cannot be given, it corresponds to the component whose frequency and wavenumber are both 0, which physically corresponds to the flow direction stripe structure with wavenumber along the span in the static state in space. The direction of $x$ is seen as a fluctuation. This is why the transformation relation given by Herbert cannot be applied to the fundamental mode instability. For the fundamental mode, the growth rate and frequency shift should be determined by directly solving the spatial instability problem, and the time-space conversion relationship formula given by Herbert cannot be used.

\subsection{Secondary instability over a Mach=6 flat plate}\label{sec:Mach6.0}

In the above subsection, the traditional Gaster transformation is proved that cannot be used to the fundamental resonance. In this section, the nonlinearity is taken into account to produce a new base flow for secondary instability analysis at Mach 6 flat plate. We use this case to further verify the correctness of our conclusions.
\begin{figure}
	\psfrag{A}[][]{$\alpha_i$} \psfrag{W}[][]{$\omega$}
	\psfrag{M}[][]{$(a)$} \psfrag{N}[][]{$(b)$}
	\psfrag{X}[][]{$x$}  \psfrag{U}[][]{$A_u$}
	\centerline{\includegraphics[scale=0.5]{photos/M6_LSTcontour_NPSE_b0.eps}}% Images in 100% size
	\caption{LST contour and NPSE at $\beta_H=0$. . (\textit{a}) LST contour, and (\textit{b}) Streamwise development of the maximum $u$-velocity disturbance amplitude at various Fourier modes by NPSE.}
	\label{fig:M6LSTcontourNPSE_b0}
\end{figure}
The freestream Mach number $\mathrm{Ma}$ of boundary layers is 6.0. The freestream temperature $T_e$ is $80 \mathrm{K}$. The wall is isothermal and the temperature $T_w$ is $520 \mathrm{K}$. 

\begin{figure}
	\psfrag{A}[][]{$A$} \psfrag{G}[][]{$\sigma_r,\gamma_r$}
	
	\psfrag{M}{$(a)$}  	\psfrag{N}{$(b)$}
	\centerline{\includegraphics[scale=0.3]{photos/M6GR_Amp.eps}}% Images in 100% size
	\caption{Secondary instability growth rate against the primary amplitude of Mack mode disturbances through the spatial SIT and temporal SIT when $\beta_2$=1.0.}
	\label{fig:M6GR_Amp}
\end{figure}
Firstly, the linear stability is performed for investigating the linear characteristic of the boundary layer. The contour of LST growth rate with various frequencies along the streamwise direction is displayed in figure \ref{fig:M6LSTcontourNPSE_b0}(a). In order to make the primary Mack mode saturated downstream to a higher enough amplitude, $\omega=1.0$ is selected here. With the initial amplitude of $1\times 10^{-3}$, NPSE method is selected to compute the nonlinear development of the primary Mack mode. The spatial distribution of the corresponding amplitudes of the peak $u$-velocity disturbance at various Fourier modes, as shown in  \ref{fig:M6LSTcontourNPSE_b0}(b). From this figure, it can be seen that the amplitude of each mode grows from the inlet $x_0=200$ and saturates after $x=440$. Then, the streamwise base flow for secondary instability analysis can be built using the amplitude evolution and shape functions results. Figure \ref{fig:M6GR_Amp} illustrates the spatial/temporal growth rates of secondary instabilities on Mack mode primary against the amplitude of primary mode (1,0). It is observed that in this case, in the earlier stage, the subharmonic mode and the detuned mode have larger growth rates and the fundamental mode dominates the secondary instability obviously when the primary mode amplitude exceeds a threshold value. This is also in accord with the conclusions in Refs. \citep{hader2016laminar,hader2017fundamental,xu2020secondary,xu2021pressureg}. 

In order to validate the correctness of the spatial growth-rate calculations, the NPSE method is employed as shown in figure \ref{fig:M6_b0_Amp_multi}. The primary mode (1,0) with the amplitude of $1\times10^{-3}$ and the fundamental disturbances (1,$\pm$1) with a tiny amplitude of $1\times10^{-8}$ are introduced at the inlet. The nonlinear evolution of these disturbances are demonstrated in figure \ref{fig:M6_b0_Amp_multi}(a) compared with the amplitude development integrated by the spatial growth rate predicted by SIT. Meanwhile, figure  \ref{fig:M6_b0_Amp_multi}(b) plots a similar nonlinear evolution except the tiny oblique disturbances  (0.5,$\pm$1) here are subharmonic disturbances. In both figures, it can be observed that the harmonic disturbances have an exponential growth downstream when the primary disturbance grows high enough. Obviously, the harmonic disturbances are excited by the primary mode disturbance. The prediction of SIT is in good agreement with the results by NPSE in the nonlinear region. It indicates that the spatial growth rates obtained by SIT are reasonable, and the SIT method is a reliable tool to analyze the secondary instability analysis. Furthermore, the stationary mode (0,1) certainly has the largest amplitude in the all harmonic spectrum, including the initial disturbances at the inlet (1,$\pm$1). This result is similar to the conclusion  in the literature \citep{liu2019fundamental}. It proves that the stationary streaky (0,1) dominates the fundamental breakdown in the secondary instability \citep{wheaton2009instability,schneider2015,chynoweth2018measurements,hader2016laminar,hader2017fundamental,xu2020secondary,xu2021pressureg}.
% (phase lock这里我觉得可以去了 这了还留着感觉意义不大 因为已经不是重点了。但是接下来你需要强调一下这里的情况和前面temporal 和linear 情况下的联系和区别。当然，主要是突出共性的地方 而差别本质上只是一个量的差别 并非质的差别。其基本规律是相似的。)
\begin{figure}
	\psfrag{U}[][]{$A_u$}
	\psfrag{A}[][]{$(a)$} \psfrag{B}[][]{$(b)$}
	\psfrag{X}[][]{$x$}
	\centerline{\includegraphics[scale=0.5]{photos/M6_b0_Amp_multi.eps}}% Images in 100% size
	\caption{Amplitude evolution of the primary mode, fundamental mode and subharmonic mode through NPSE and SIA. (\textit{a}) lines - fundamental mode, symbols ($\blacksquare$)- SIA, and (\textit{b}) lines - subhamonic mode , symbols ($\blacksquare$)- SIA.}
	\label{fig:M6_b0_Amp_multi}
\end{figure}
% 这里应该之前有个帽子，说一下调查detunded parameter $\epsilon$的意义。

In the traditional way, the classical Gaster transformation has been employed and the results are displayed in figure \ref{fig:M6GR_Amp_phase}. From the figure, a similar conclusion can be drawn. In subharmonic resonance, the growth-rate of the temporal mode after Gaster transformation is in good agreement with the growth rate of the exact spatial mode, which is in consistent with the traditional cognition. However, in the fundamental resonance, the growth-rate of the temporal mode after Gaster transformation is very different from the growth rate of the exact spatial mode, especially when the primary amplitude $A$ is large. This is not in line with the traditional cognition. 

In the spatial analysis, this view will be discussed considering the nonlinear effect. The Fourier analysis is employed on the eigenfunctions of the two types of secondary instabilities. Figure \ref{fig:M6_Fourier} shows the Fourier analysis results of the fundamental and subharmonic eigenfunctions. The amplitudes of the primary Mack mode disturbances are $A_{(1,0)}$=0.01 and $A_{(1,0)}$=0.15, respectively. It can be known that the spectrum (0, 1) is always the largest one in the spectrum space for the fundamental mode. It confirms that the stationary streak (0, 1) is always the dominant structure in the fundamental breakdown\citep{wheaton2009instability,schneider2015,chynoweth2018measurements,hader2016laminar,hader2017fundamental,xu2020secondary,xu2021pressureg}. By the contrary, there are no stationary structures in subharmonic modes, and a subharmonic travelling disturbance (0.5, 1) dominates the subharmonic breakdown. When the component of $m$=0 is relatively large, the traditional Gaster transformation can no longer be used.

\begin{figure}
	\psfrag{A}[][]{$A$} \psfrag{G}[][]{$\sigma_r,\gamma_r$}
	
	\psfrag{M}{$(a)$}  	\psfrag{N}{$(b)$}
	\centerline{\includegraphics[scale=0.6]{photos/M6GR_Amp_phase.eps}}% Images in 100% size
	\caption{Comparison of the secondary instability growth rate against the primary amplitude of Mack mode disturbances through the spatial SIT and Gaster transformation when $\beta_2$=1.0. (\textit{a}) fundamental resonance. and (\textit{b}) subharmonic resonance.}
	\label{fig:M6GR_Amp_phase}
\end{figure}

\begin{figure}
	\psfrag{U}[][]{$|u'|/|u'|_{max}$} \psfrag{Y}[][]{$y$} \psfrag{m}{$m$}
	\psfrag{A}[][]{$(a)$} \psfrag{B}[][]{$(b)$}
	\psfrag{C}[][]{$(c)$} \psfrag{D}[][]{$(d)$}
	\centerline{\includegraphics[scale=0.4]{photos/M6_Fourier.eps}}% Images in 100% size
	\caption{Fourier analysis of secondary instability eigenfunctions. (\textit{a}) fundamental mode when $A_u=0.01$, (\textit{b}) fundamental mode when $A_u=0.15$, (\textit{c}) subharmonic mode when $A_u=0.01$ and (\textit{d}) subharmonic mode when $A_u=0.15$.}
	\label{fig:M6_Fourier}
\end{figure}

\section{Conclusions and future work}\label{sec:Conclusion}


In this study, in the secondary instability analysis of the Mack mode of the 4.5 Mach plate, the phenomenon that the traditional Gaster transformation cannot be applied is found, and then the reason for this limitation is derived mathematically in details. Finally, the effects of nonlinearity and non-parallelism are considered over a Mach 6 flat plate, which further verified that our analysis is reasonable and correct. In summary, the traditional Gaster transform is not applicable to the fundamental secondary instability problem with large primary amplitude. Please refer to section\ref{sec:mathematical} for the standard spatial mode calculations.

\section{Acknowledgement}
This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. G2021KY05101) and the National Natural Science Foundation of China (Grant No. 91952202). The authors are very thankful to Professor Xuesong Wu from Imperial College London for his useful discussions and suggestions. Dr. Cunbo Zhang from Institute of Applied Physics and Computational Mathematics in China and Junqiang Bai from Northwestern Polytechnical University are also appreciated for their help.

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